help-28

[topaz saddle]

how am I meant to do this? I’m trying to use the formula but what do I do when there’s a 0?

[topaz saddle]

This is the question I’m trying to solve if ur curious

[topaz saddle]

My notes are on the left, that I’m trying to use, and my working out on the question is on the right

[inland moth]

why did your P(Y = y) suddenly all become 1/3

[topaz saddle]

Im trying to do what my notes r on the left

Fir finding the mean

If its wrong mb idk what im doing

[inland moth]

ok but in your problem the probabilities are the ones you worked out above

then you just multiply and add them

doesnt matter if it has a 0

just do it

[topaz saddle]

The ones in orange?

[inland moth]

yes

[topaz saddle]

Ok give me a bit

[topaz saddle]

Wdym multiply and add them?

[inland moth]

as the formula states

xP(X = x)

[topaz saddle]

Like this?

[inland moth]

yea

[topaz saddle]

Did I get it right

[inland moth]

you missed 3

[topaz saddle]

oh wait

oh my brain was still on part (b) so i skipped 3

awetgfaerg

[inland moth]

also small thing in your scratch work

you are multiplying with fractions

not adding

no need for a common denom

just multiply it with the numerator

[topaz saddle]

Correct?

I don’t think I did it right

[topaz saddle]

What

[inland moth]

i dont think P(Y = 2) is 8/12

[void nova]

Indeed

[topaz saddle]

Am I meant to use this

[void nova]

You've correctly calculated the orange values but then why didn't you use them properly? @topaz saddle

[void nova]

Huh? This is correct, though, yes

[topaz saddle]

shioshi said 8/12 isnt right tho

[void nova]

In fact it is not

Where did you get 8/12 from?

[topaz saddle]

wait

fml

idk why i wrote

8/12

ur right

ill do it again smh

Idk

[inland moth]

????

[void nova]

Why did 2 and 3 become 2/36 and 3/36???? (green part)

[inland moth]

do you need an electron microscope?

[void nova]

By the way, why aren't you using the ORANGE values in the E(y) calculation?

[topaz saddle]

Ok mb I’m tired

[void nova]

Also the correct notation is E(Y), not E(y)

[topaz saddle]

i got 19/9

this has to be right

[inland moth]

good.

[topaz saddle]

also i make 2/36 so it can have the same denominator no?

so i multiply the top

[void nova]

Why??

[inland moth]

like i said before

[void nova]

That's not at all how multiplication of fractions work ...

[inland moth]

same denominator is only required when you are ADDING/SUBTRACTING

multiplying is not

just multiply with the top of the numerator

done deal

[void nova]

Definitely need to revise BASIC FRACTIONS ALGEBRA @topaz saddle

[inland moth]

sorry for caps just wanted to emphasize that

[topaz saddle]

bruh im just getting yelled at

😔

[void nova]

That's only good for you, when we forget something it is obvious that there's a need to revise the topic 🤷‍♂️ this is how you can get better

[topaz saddle]

ok well im working on it

thanks for the help you two

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[cold wren]

Can someone help me with this?

[cold wren]

nvm

i got it

.close

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[sacred yarrow]

👀

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[slender finch]

help on Cii pls

struggling to figure out how to parameterise the surface

so I can dot the curl with the unit norma vector

i have calculated the curl of V but i am not sure how to find the unit normal vector since i would need to do the partials of my parameterisation with respect to their variables crossed producted

<@&286206848099549185>

chatttttt

helppp me por favorrr

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@slender finch Has your question been resolved?

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@slender finch Has your question been resolved?

[slender finch]

@floral scroll

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[twin willow]

how do i do $\int cos^4(x)dx$?

[glossy valve]

Kiyotaka

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What step are you on?
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2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[willow sedge]

many ways to go about it
compound angle formula (indefinite)
walli's formula (definite)

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[hollow moth]

I need help integrating this expression with respect to x:

[hollow moth]

im just confused because i see dx in the denominator, so i am thinking of doing ln(x), but since they are derivatives, im not sure i can do that

like I think the answer is da * ln(x) / dy

[fathom saddle]

d²a/dxdy would make sense to me, but this looks broken as given

What's the context?

[hollow moth]

so honestly, i am doing an astrophysics problem. I am not even sure if i am on the right path for the astrophysics problem, but I have this equation (I solved for Iv), and I want to integrate with respect to omega.

But since this is kind of overcomplicating things, I just thought it would be easier to try and integrate the original problem.

[fathom saddle]

Oh god it's physics notation

[hollow moth]

sorry, i was trying to keep it math related since ik it's a math server

[hollow moth]

wait is d²a/dxdy the answer?

[fathom saddle]

Haha that's fair. I wish I knew some astrophysics

[lunar veldt]

no

[rose zephyr]

No I believe Kaynex meant that if the expression was d^2 a / (dx dy) then it would make more sense

[lunar veldt]

i would say da/dy +C(y)

[lunar veldt]

if you differentiate this with respect to x , it gives d^2 a/dxdy because C(y)=0 , it only depends about y

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[winged elm]

Hi

[winged elm]

I need help with this

how can I know the height of the triangle

[weak bolt]

3² = 2² + DM²
DM² = 3² - 2²

$DM = \sqrt{3^2 - 2^2}$

[glossy valve]

╰ 𝕃 𝕌 ℂ 𝕀 𝔽 𝔼 ℝ ╮

[winged elm]

That 3 is just the size of the mass

[weak bolt]

3m

3 meter?

[winged elm]

No, its 3 mass units

I only know the length of the base

which is 4a

And the length of the base of the right angle if I cut it in half is 2a

Here is a solution to a similar question in the textbook

It just assumes that the the altitude from the right angle to hte hypotenuse is 2a?

I don't understand why

Is there something about equilateral triangles that I'm missing here

isoceles*

[weak bolt]

U have position of A and B, and u need to find out position of C

Let M in middle of AB
So | (Xb - Xa)/2 , (Yb - Ya)/2 |is position of M
Then calcul Distance CM

@winged elm

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[clear loom]

Anybody know what type of math this is and how i can learn to do it

[subtle hatch]

Get any high school textbook. It's basic geometry.

Or youtube

[clear loom]

I just need a name

when i search up line

it doesnt pop up and all i see is lines

[subtle hatch]

What grade u in

[clear loom]

9

[subtle hatch]

Get ur 9th grade math textbook

[clear loom]

Ok

.close

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[hollow moth]

I have the equation v * (dλ/dv). If λ = c/v, does that mean (dλ/dv) = -c/v^2? So then the equation becomes v * (-c/v^2)?

[hollow moth]

did i do that correctly?

[lime ether]

v * d lambda/dv isn’t an equation

[hollow moth]

wait im confused. is the operator dλ/dv what makes it invalid?

[lime ether]

$v \dv{\lambda}{v}$ is not an equation

[hollow moth]

sorry it's been a while since ive done calculus

[glossy valve]

knief

[lime ether]

and that’s not an operator

[quick summit]

Could anyone help me with this one

[hollow moth]

so to make it an equation, do I need to have another equation and an '=' sign?

basically set 2 equations equal to each other?

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Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[hollow moth]

or am i lost entirely

[lime ether]

not another equation, you have no equation at all

you need an = sign

did you mean

$v = \dv{\lambda}{v}$

[glossy valve]

knief

[hollow moth]

okay i think i understand. So I need an equation for dλ/dv. Then once i have an equation (such as v = dλ/dv), do i then cross multiply and take the derivatives respectively?

[lime ether]

what are you solving for? lambda?

are you looking for a solution

[hollow moth]

the original problem is that I need to prove v * f_v = λf_λ.
In lecture, we talked about how f_v = (dλ/dv) * F_λ.

So I am starting with vf_v = v*(dλ/dv) F_λ

sorry the formatting got messed up because of discord. the term is f_λ, not f*λ

[lime ether]

$v f_v = \lambda f_{\lambda}$

[glossy valve]

knief

[lime ether]

this?

[hollow moth]

yes

sorry, i don't know latex lol

[lime ether]

what is f_v

and f_lambda

[hollow moth]

i'm not sure i was ever given formulas for them. I kinda just assumed i was supposed to use that hint we discussed in lecture. Is it not enough information? Cuz then i'll just do the problem at office hours

[lime ether]

is there more context?

[lime ether]

can you explain what they are though?

it’s notation for something

what is it notation for

[hollow moth]

They're both measurements for the specific flux. They just use different units

[lime ether]

i’m not physics savvy, are they functions or constants?

[hollow moth]

function. I found this formula in the textbook:

but I dont see anything for Fλ.

[lime ether]

do you have the original problem by chance?

[hollow moth]

part b is completely unrelated

[lime ether]

what subject is this for

physics yea but like

[hollow moth]

astrophysics

[lime ether]

https://physics.stackexchange.com/questions/725928/converting-between-f-nu-and-f-lambda-spectral-density

[hollow moth]

im sorry if im taking up so much of your time, i thought this problem was from my lack of calculus understanding, but if it has more to do with astrophyiscs, ill do it at office hours

[lime ether]

i think i found your question

[hollow moth]

thank you so much. Sorry i took so much time of ur time, i just thought it was because of my lack of calculus understanding.

[lime ether]

yea i don’t have the physics knowledge to even understand the question

[hollow moth]

.close

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[tropic lion]

How to do question 8

I forgot how to find the vertex

[gritty rose]

,rccw

[gritty rose]

Do you know the vertex of |x|

[tropic lion]

No

😭

[gritty rose]

,tex .abs def

[glossy valve]

riemann

[gritty rose]

Plot that and see if you can find the vertex

[tropic lion]

Emmmmm

Huh

?

Plug 0?

What’s the method of finding vertex, is to find the domain?

@gritty rose 😭helppp

[pearl pilot]

the vertex of |x| is at the origin

[tropic lion]

Enmmm

I don’t understand 🥲

[tropic lion]

Oh yeah

So how to do it?

Oh

Make it zero. So x=-4

[pearl pilot]

ya

and y=?

[tropic lion]

3

[pearl pilot]

gg

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[late flame]

YOLO need help solving this rhombus

[late flame]

Is it 63

And help on this

E

<@&286206848099549185>

[robust slate]

Yes

[late flame]

Ye

[robust slate]

Recall that the diagonals of a rhombus are angle bisectors

For example, DB bisects angle ADC

[late flame]

Yes

[robust slate]

So think of how you can work backwards to get ADE

[late flame]

Ihhh

Do I like subtract???

[late flame]

Oh yeah the diagram resets after each problem

Will it still be 27

[robust slate]

What does the angle bisector tell you about angle ADE

[late flame]

Idk

[robust slate]

Do you remember what an angle bisector is?

[late flame]

Yes it’s the thingy that bisects angles into two half’s

Equal

[robust slate]

So … ADE is one of those two “equal angles”

So what is ADE half of

[late flame]

ADC

[robust slate]

So find ADC

And then you’re basically done

[late flame]

How would I find adc??

[robust slate]

• what are you given
• how does it relate to ADC

[late flame]

I was given 140 for DAB

but how do I like connect the dots to adc

[robust slate]

You should notice that they’re adjacent angles

Take that as you will

Further hint: ||think of your angle relationships from parallel lines||

[late flame]

Ahhh

Sooo supplementary???

OHHH

ADC equals 80

YEAH???

And the half of that is 40

[robust slate]

Um

Check your arithmetic

[late flame]

Yes 😀

Huh

[robust slate]

How is ADC = 80

[late flame]

180?

-140

😀🥲

OH WAIR BO

ITS 60

😂😂😂

[robust slate]

How is 180-140=60

[late flame]

40?

[robust slate]

Yeah ._.

[late flame]

Yeah…I’m losing sleep

Help with this?

🙏

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[late flame]

.close

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[neat terrace]

I dont understand why for part a

we multiply the x value by 2

but then in part b

we compress it by 4

how dyu know when it should be compressed or stretched out

[storm lintel]

yes

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@neat terrace Has your question been resolved?

[wicked stream]

when u had to sketch graph of f(x) from f(2x) you stretch out
when u have to sketch graph of f(4x) from f(2x) you compress

making f(x) from f(2x) means
f(x) attains values 1/2 the speed of f(2x)

hence stretched

where in 4x its twice the speed of 2x

hence compress

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[solemn sand]

I didn't understand the question as well as the logic used for it

[onyx glen]

ok let's see

do you know what "ordinary pack of 52 cards" refers to

[solemn sand]

Yes

[onyx glen]

ok

so you know that in a single draw the probability of getting a heart is 1/4, yes?

[solemn sand]

Yes

[onyx glen]

and since we put the card back in, that never changes

[solemn sand]

Yes

[onyx glen]

and we want to look at the probability of getting at least one heart over multiple draws

so for example what would that be if we draw 3 times?

[solemn sand]

So p(x>=1)

[solemn sand]

1-27/64?

[onyx glen]

1 - (3/4)^3, yes

ok

so if we do n draws then what will be our chance of getting at least one heart

[solemn sand]

Will increase

[onyx glen]

... not its behavior

its actual exact value

of course it will increase

[solemn sand]

Right? Cause it'll just be 1-(3/4)^n n= number of cards we have picked and put back?

[onyx glen]

1 - (3/4)^n, yes

i did say n draws

ok

now your questions are

find the smallest n for which that >= 1/2

and find the smallest n for which that >= 3/4

[solemn sand]

Ohh

That makes sense

But I still don't get how 'there is atleast an even chance of drawing a heart' makes n>=1/2

[onyx glen]

"even chance" means one half probability

[solemn sand]

Ohhhh

[onyx glen]

it is a somewhat old fashioned wording

[solemn sand]

And after this what do we do?

[onyx glen]

after what

[solemn sand]

So this is incorrect?

[solemn sand]

Figuring out What the questions mean

[solemn sand]

For this we need to find n in (3/4)^n=1/2 and next one (3/4)^n=1/4?

[onyx glen]

it is correct but phrased in a different way and (independently) a bit overcomplicated

[onyx glen]

<= on both, not =

[solemn sand]

Oh I forgot

So the answer will be in <=?

[onyx glen]

...

no,

after solving the inequalities, you need to find the smallest natural n which satisfies them

and those smallest n's are your answer

[solemn sand]

Solve this by using log?

[onyx glen]

some logarithms will happen yeah

[solemn sand]

For even change it'll be n=3

Right?

[onyx glen]

yes

[solemn sand]

And 2nd one n=5?

Btw

How do you understand these questions

Like what's your train of thought

For probability questions permutations and combinations those types of stuff

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[wheat swallow]

Can anyone verify if this is correct?

[wheat swallow]

If you keep following the predecessor it has to contain the 0 vertex at some point right?

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[twin wolf]

im not sure how to start

[torn jolt]

What does u.v=0 imply?

[modest obsidian]

first understand what the problem gave u

same thing

[twin wolf]

that u and v are perpendicular if their dot product is 0

[torn jolt]

Yes

Can any two orthogonal vectors span R²?

[twin wolf]

sorry we didnt learn what span is yet

[torn jolt]

Alright

Say you have (0,1) and (1,0) column vectors

Can you express any vector w = (a,b) in terms of them in R²?

[twin wolf]

is this the

e1 e2 stuff?

[torn jolt]

Yes

[twin wolf]

we can use linear combination?

[torn jolt]

Yep

[twin wolf]

so if we let vector w = <x,y>

then w = xe1 + ye2?

[torn jolt]

Yes and e1=u and e2=v because both u and v will "generate" R² like (0,1) and (1,0) do

For example just consider u=(0,1) and v=(1,0)

[twin wolf]

so how can we use this to help us?

[torn jolt]

so w=xu + yv

proj_u(w)=?

Take w=(1,1)

w = 1(1,0) + 1(0,1)

proj_(1,0)(w) = ? (the projection vector you get after projecting w onto (1,0))

[twin wolf]

,rccw

[twin wolf]

this what i have so far

[torn jolt]

Indeed. Try to simplify it a bit using u.v=0

[twin wolf]

alright

hold on

[torn jolt]

Sure np

[worn matrix]

the handwriting is so ass beam 💔🥀

[twin wolf]

stfu

[twin wolf]

okay im back srry was helping a friend

im sending image

idk if my i did any illegal algebra so llmk

[torn jolt]

All legal

[twin wolf]

great

[torn jolt]

So it's xu

[twin wolf]

am i allowed to cancel u dot u on top and bottom

[torn jolt]

Yes because they're scalars

[twin wolf]

oh right

[torn jolt]

Similarly, proj_v(w) = yv

[twin wolf]

yea

[torn jolt]

With this information at hand, what options can you eliminate from here?

[twin wolf]

i think option a is not always true

[torn jolt]

this info is not needed nvm

[twin wolf]

yes

[torn jolt]

So what's (xu).(yv) ?

[twin wolf]

xy(u dot v)

xy*0 = 0 then

o

[torn jolt]

Yep

[twin wolf]

so then option A is true

[torn jolt]

Indeed

[twin wolf]

option B is going to be w = xu + yv

isnt that just linear combination thats what we said was true right

[torn jolt]

Indeed

To see why you need to prove that u,v will "generate" (or span) the entire R² when u.v = 0 (equivalently you can express any vector in R² in terms of u,v)

You'll hopefully see this when you'll do span

[twin wolf]

ah okay

so option C is saying that xu + yv = xu OR xu + yv = yv

i cant see how that will be true

[torn jolt]

Yep not always true

[twin wolf]

but last option if xu = 0 then yv = xu + yv is true

so option C is the correct answer

[torn jolt]

Indeed

[twin wolf]

ohhh ok i get it now thank you 🙏

[torn jolt]

Np. Have a great day

[twin wolf]

u too

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[waxen cliff]

I'm not sure how to start (12(a))

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[waxen cliff]

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[trim umbra]

hi guys so i was trying to solve a quadratic function with Bhaskara's formula.

the function is

[trim umbra]

and

the resutls i got from it are (-1,0) and (3,0)

[faint iron]

The quadratic?

[onyx glen]

Bhaskara's formula? & what are you actually solving? the equation f(x)=0?

[rain merlin]

bhaskaras formula is just quadratic formula iirc

[errant elbow]

its another name for the quadratic formula

[onyx glen]

that so?

ok

[rain merlin]

brazilian acc to a google search

[trim umbra]

and so i put the coordinates on desmond to keep better track of how it was coming togheter

[faint iron]

3 does not work

[trim umbra]

but when putting the formula it was off

shouldnt it go exactly there?

[faint iron]

You did something incorrect

[errant elbow]

your substitution might be incorrect

[rain merlin]

i dont think your roots are right tho

[rain merlin]

yeah can you show your working for this

like how exactly did you go about and get the coordinates you stated

[trim umbra]

ok

so first i naturally replaced the values with the corresponding ones of my function (sorry for using paint to do this)

[amber robin]

b is -4
-b is 4

[trim umbra]

oh f god

gimme a moment i need to quickly try that

guys i think i fd up again

[trim umbra]

isnt the formula incomplete?

here shouldnt it go with *a

[normal light]

Yes, but I divided both sides by -2

Just to clarify, it's -2x^2 - 4x + 6 right

for f(x)

So if f(x)=0, then -2x^2-4x+6=0

If you divide both sides by -2, we get x^2+2x-3=0

I used the divided equation for the formula rather than the original

[trim umbra]

i think i get most of it but where has the -2 gone?

nvm it became invisible since its a 1 now, rihgt?

[normal light]

I don't think the -2 matters once you've divided both sides by it

And if you use the original, it still wouldn't be a problem

[normal light]

For this, you have (4+8)/-4 or (4-8)/-4

So 12/-4 = -3, -4/-4 = -1, which is identical to the answers I got

[rain merlin]

your simplification is incorrect

your fractions are correct

[trim umbra]

i get it now, and it does match the roots that desmos says tysm

[normal light]

No problem

[trim umbra]

the -(-4) had me tripping

i forgot it was just 4 positive

[rain merlin]

good job

[normal light]

It's okay, it takes time. If it helps, I usually just directly turn it to +4 for the sake of less brackets

[trim umbra]

tysm yall i am actually passing the test today ^^

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[torn jolt]

Hey, I'm solving some differential equations using the method of undetermined equations and I stumbled upon a problem where the particular solution is a product of a polynomial and an exponential function, I'm not sure how to get the Particlar solution, help is appreciated.

[amber robin]

Do you know how to solve differential equations like these in general?

[torn jolt]

yes but Ive never solved one where f(x) is a product of a poly and an expo

[amber robin]

What method do u follow?

[torn jolt]

the method of undetermined coeffs

[amber robin]

Hmm

How about taking (ax² + bx + c)e^4x?

[torn jolt]

that's what I initially did but I got an unsolvable system for the 3 constants

a, b and c

[amber robin]

Interesting

Cause I got a = 1/2, b = -15/16 and c = -5/32

Oh, wait

Nvm

,w 715/162-9/2-4/9

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[torn jolt]

is that c?

[amber robin]

That ÷ 7 is c

a = 4/9
b = -143/162
c = -43/567

This is what I got

,w y'' + 2y' - 6y = (8x^2-7x-9)e^(4x)

[glossy valve]

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[amber robin]

The a and b values seem to be correct

But c is -43/729

I got -43/567

[torn jolt]

hmmm

[amber robin]

And then there's the constant terms

Which idk how to get

,w 2a + 8b + 16c + 2b + 4c - 6c

[glossy valve]

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[amber robin]

That's correct

Hmm

[torn jolt]

it is indeed correct

Thanks man!!

I really appreciate it

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[simple sandal]

i have a really general question im typing this in my ti84 and i keep getting dimension error could someone help me

[simple sandal]

im doing matrix a as 4x2 with u and v and then matrix b as 4x1 with w and then doing rref(augment([A],[B])) and keep getting dimension error

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[umbral dome]

in the ti-84 RREF can only be done if there are at least as many columns as there are rows (for some reason)

[steel solar]

Could you send a picture? Also, did you get your question resolved?

[simple sandal]

yeah got it resolved

[umbral dome]

you can get around this by augmenting additional 0 columns

[simple sandal]

yeah i rmb my prof mentioning this and i just found it in my notes

thanks a ton

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[gritty rose]

do you mean rho(2) depends on rho(1) ?

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[urban steeple]

Can someone look at my proof and tell me if it’s valid and give feedback if possible, this is a proof for the associative identity using the Peano axioms and the propositions in the Analysis I book by Terence Tao this is my first time trying to do a real analysis proof (I’m technically in high school but self teaching) so bare with me if it’s not the greatest 😭🙏

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[sturdy valve]

can someone help in part b

like i created too many variables and could not proceed

[atomic hare]

we are assuming the refractive index of the "air" is 1?

[sturdy valve]

yep did till there

yes we are

[sturdy valve]

then after this i dropped a perpendicular from that pt (blue yellow intersect) to the ground

or did i do the first step itself wrong

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@sturdy valve Has your question been resolved?

[sturdy vine]

Hi

[gritty rose]

why do you think light ray is perpendicular to the ground after it crosses the transparent liquid

[sturdy valve]

nono

i didnt say the light ray is perpendicular

i just drew a perpendicular

imaginary line

for calculations with angles

[gritty rose]

oh i see

[sturdy valve]

yeah

any idea what i should do

[gritty rose]

you should be able to figure out the angle of incidence at the blue line given refrac index = sqrt(3)

and snell's law

[sturdy valve]

oh and from there i can get refraction angle

i will try

in sometime..i moved on to another question give few mins

@gritty rose im not able to find angle of indicence 😔

incidence

[gritty rose]

did you try to use snell's law

and did you find the angle the light ray makes with the liquid

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@sturdy valve Has your question been resolved?

[sturdy valve]

yes

[gritty rose]

and why are you stuck

[sturdy valve]

the angle of refraction?

[sturdy valve]

sin i = root 3 sin r

i have 2 unknowns now

[gritty rose]

the angles in this right triangle

[sturdy valve]

yeah no i wasnt able to find it

[gritty rose]

you also need to find this tangent line in terms of the angles as well

then you'll have two equations and two unknowns

[sturdy valve]

oh ok so i should find the angle that makes with the ground?

ykw im done with this question for now

ty for helping

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[high ridge]

"is the orthogonal projection of v on n" what about the orthogonal projection of n on v? I'm confused because the scalar product is meant to be commutative

[clear pumice]

n on v would just be the case where v is the unit vector instead of n

(yes scalar product is commutative so flipping it will still give you the original projection)

but flipping which one is a unit vector changes the projection

[rapid rain]

also this is slightly wrong

$( \mathbf{\hat n} \cdot \mathbf{v}) \mathbf{\hat n}$ is the orthogonal projection

[glossy valve]

rafilou is not not born in 2003

[rapid rain]

$\mathbf{\hat n} \cdot \mathbf{v}$ is only the coefficient of the orthogonal projection

[glossy valve]

rafilou is not not born in 2003

[umbral dome]

sometimes people call the coefficient of the projection the "scalar projection"

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[supple plover]

the first image is my notes and what ive done so far for this and previous questions, I am not sure if everything is right, nonetheless, I am stuck. I am not sure what my professor means of 'What identity do we have for sin2alpha'

and im stuck on how to rewrite the area of the isoscles triangle in terms of r and theta

if this is confusing lmk, ill do my best to explain again. I am just lost about all of it entirely haha

[devout valley]

(have you heard of the double angle formulas? or the addition ones, how to expand something like sin(A + B)? I presume that's what they're after)

[supple plover]

the sin(2alpha) = 2sinalpha cosalpha

that?

to be honest no... my previous prof skipped that whole chapter so I have never done them before..

😭

[devout valley]

As for rewriting, isn't that just a matter of using theta = 2alpha in the expression for the triangle area you found above it?

[devout valley]

Damn that's annoying

[supple plover]

yea my professor skipped A LOT so im very frustrayed they said, 'its not that important to know going into calc, just extra stuff'

[supple plover]

sooo like writing it as, sin(2alpha)= 2sinalpha cosalpha=sin theta?

wait idk

AH

[devout valley]

That's not great esp when it's likely they'll use things you're assumed to have learned before in multiple areas

[clear pumice]

i feel like you use trig identities for like half of calc 2 🙏

[supple plover]

bro.. welp guess im going to have to go back and teach myself 😭 😭

[devout valley]

(and you can use that as part of it sure, but also as alpha = theta / 2, that should then give you a nice way to find the area of the triangle too )

in other words, putting alpha = theta/2 here (and then of course you can double angle simplify stuff too!)

[supple plover]

im sorry if its too much trouble, but is there a way you could draw what you're saying (if that makes sense) onto the image. Because reading everything rn i feel like my head abouta explode 😅

if it is, you dont gotta <3 i can figure it out LOL

[devout valley]

What I was trying to say is that you can say
[
\frac12 (2r\sin(\alpha))(r\cos(\alpha)) = \frac12 \qty(2r\sin(\frac{\theta}2))\qty(r\cos(\frac{\theta}2))
]
(and of course, that right hand side is still simplifiable)

[glossy valve]

@devout valley

[devout valley]

The right hand side is the area for A in terms of only r and theta, though not in its simplest form

[supple plover]

ohhhhh okay okay i see

so then to simplify I would times it by its reciprocal right?

oh wait

wouldnt the 1/2 and the 2 cancel out?

[devout valley]

They can, but don't cancel them out for now

[devout valley]

[supple plover]

oh okok

wait.. are you hinting that thats the simpliest form 👀

[devout valley]

(or part of it )

[supple plover]

OHHH

so with that equation i can divide it by 2 then to simplify it even more????

OMMG 💅 my gears are back online

[devout valley]

Or more what I'm hinting at, you can factor both of the $r$'s out of here, then you get
[
\frac12 r^2 {\color{green} \qty(2 \sin(\frac{\theta}2) \cos(\frac{\theta}2)) }
]

[glossy valve]

@devout valley

[devout valley]

It should hopefully be clear what the green turns into, you already got it

[supple plover]

i think i do, because the green was what i typed earlier right?

yeyeye then i do

idk if you lookes at the other guided questions, but the steps you and I did together, does that answer those questions? or are there more to this problem? that im missing

i think so... but i just wanted to make sure

[devout valley]

For the ones before, do you have like the full prompt from here? I feel that for example...

...here may not be the answer they're looking for, though you're not "wrong" I would say

I'm personally happy with them, pretending you were doing them for me

[supple plover]

would you like for me to send you the full image?...

[devout valley]

Yes please

[supple plover]

okok i got you <3

[devout valley]

Ah I see, I think you're fine in that case seems like the guiding questions are more like supposed to lead you to answer the "main questions", I think

[supple plover]

yea im starting to think, thats the case to

im sorry if im taking up your time but i had a question for number 5?

[devout valley]

You aren't taking my time up, I'm chilling and eating and would be happy to help

[supple plover]

youre awesome

so when it says n is it the number of sides in the polygon?

in this case would it be 8?

[devout valley]

Yep, in the pictured one it would be 8, but it can be whatever in general

[supple plover]

would I use the equation we just found for number 5

or is this its own thing

[devout valley]

You can, and very much should, make use of that, yep

[devout valley]

(look at the picture I think you know what to do )

[supple plover]

okok, i believe im comfortable enough to do this one of my own...thank you so much for the help ❤️

[devout valley]

Awwww, it's always a pleasure to help you see ya around soon

[supple plover]

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[supple plover]

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✅

[supple plover]

ahhh im sorry, i realized something and im now confused

for the equation in what i wrote sin(2alpha)= 2sinalpha cosalpha=sin theta?

would the alpha be theta instead?

oh ok

[devout valley]

Gonna interrupt you

[supple plover]

oop

go right ahead

[devout valley]

You know both that $2\sin(\alpha)\cos(\alpha) = \sin(2\alpha)$, and also that $\theta = 2\alpha$, which is presumably the way you got there

[glossy valve]

@devout valley

[supple plover]

mhmhm

[devout valley]

So then you could of course plug that in

[devout valley]

That's just saying this in a different order if that helps?

[supple plover]

yeaa so, the equation is 1/2r^2 (2sin alpha(cos alpha))?.....

[devout valley]

Yep, that's one form of the equation, then of course as above, you can cook more with that and get something nice, in terms of only r and theta as they asked

[supple plover]

is that when we cancel out of 1/2 and 2?

[devout valley]

Not like that

[supple plover]

oh

[devout valley]

notice inside the here, you have...

[devout valley]

...the left hand side of this

[supple plover]

OH it simplifyes to the sin(2alpha)

[devout valley]

So you can simplify the formula to (1/2)r^2 sin(theta)

[supple plover]

oh yea it would be theta not alpha

because the 2alpha turns into theta

[devout valley]

Yep that's it

[supple plover]

how would I answer the 'what part of this identity is present in the equation we have' guided question for #4? I am confused on what they mean by part?

unless im being slow

[devout valley]

Basically how you manipulated it

[devout valley]

The green (you can replace the theta/2's with alphas of course)

[supple plover]

ohhh i see what u sayin

[devout valley]

You can basically say something like "oh look, we have (1/2)r^2 * (2sin(a)cos(a)), but oh shit, that (2sin(a)cos(a)) is in the identity 2sin(a)cos(a) = sin(2a), so we can use that" (of course, a bit more formally worded of course )

[supple plover]

ong thats what im doing right now 😭

im like of whoopy di do, this turns into this

[devout valley]

Yep it is

[supple plover]

chart im not even kidding, they want this due tonight and im question 4 out of 10. I am so grateful to you rn for helping haha

[devout valley]

Awwwww that's quite mean though, esp if the questions are this long

[devout valley]

Are the questions the numbered bits here, you mean?

[supple plover]

they said it shouldnt take more than hour if you knew what you were doing

[supple plover]

yeaa

these are the next of them

[devout valley]

Ohhh, I was about to think they were going to set you like multiple of the whole set of questions, I'd have died if it were like that

[supple plover]

LMAO OH

[devout valley]

And well, we definitely will get them done by the end of the day at least (assuming you have the time)

[supple plover]

yup I do! I canceled plans to get this done and im PST so I have a bit before its 11:59

[devout valley]

Awww yea you'll be alright then

We can walk through as many of these as we can (and I think we can manage them all!)

[supple plover]

OMG YOURE AWESOME

so for 5, when they ask in the guided questions how many triangles are there. Are they refering to like the sides?

ima try to draw at what im refering to

oh wait

the pink line

[devout valley]

What about it? (I think you see where you're meant to go!)

[supple plover]

okok

[devout valley]

[they're basically asking you how many of these you have ]

[supple plover]

AhAH okok

btw am i allowed to keep this chat open for however long?

[devout valley]

Pretty much yep, there isn't like a time limit, basically until you're done with the questions

[supple plover]

ohh i see

thank you

for when they ask what the area of the shape is, do they mean an equation or a value? And if its a value wouldn't it just be what n is equal to?

[devout valley]

Well, I mean, the area will be an expression that depends on your variables n, r and theta

It won't be just n though, if you think about it for a moment-

[devout valley]

You know the area of each of these triangles, we just found it, they have area (1/2)r^2 sin(theta), and you can make that polygon with multiple copies of those triangles

How many triangles do we have?

[supple plover]

for the n=4 one we have 4

triangles

[devout valley]

What if n was 5? 13? 69?

[supple plover]

we have 5 triangles, 13 triangles and 69 triangles 😏

so whatever n is, is how many triangles?

[devout valley]

Yep I was trying to get you to say "there are n triangles"

[supple plover]

ohhhhhh

LOL

so then would we implement the n into the equation we just found?

if implement is the right word haha, my brain

[devout valley]

I'll let you have that one

So, now, we have n triangles, each of them having area (1/2)r^2 sin(theta) each, so what's the total area?

[supple plover]

😀

uhhh

[devout valley]

It's an easy question

[supple plover]

it is? am i just overthinking it? lOL

[devout valley]

You probably are which is why I said it's easy

[supple plover]

is the answer in this text

[devout valley]

It isn't yet, but it's basically one step to get it

"if I have two cats that love to stalk me, and each of them are 4kg each, what is the total weight of cats I have to carry?"

[supple plover]

the total weight of cats is 8kg each

would i just double the n value?

oh

[devout valley]

Let me change my question a little bit again, what if instead now I had four cats, still 4kg each?

[supple plover]

well it would be 16kg in total for all four cats

[devout valley]

The triangles are the cats, the triangle areas are the cat weights

[supple plover]

so we have 4 triangles, the triangle areas is the equation we found earlier ( the (1/2) r^2 sin theta)) ?

would we then be essentially adding the triangle area equation thingy mabob together??

[devout valley]

Basically yep, by however many triangles you have

Of course, in the general case, replace 4 with n

[devout valley]

If I had n cats of mass 4kg, then I’d have to carry (4n) kg of cats, who would probably be very amused (and waiting for their snuggles )

If their mass was instead (1/2)r^2 sin(theta), I think they’d be looking at me as to why they have such a weird mass, and wondering if I’m trying to use them to explain a problem to someone

[supple plover]

LMFAO

reading this problems makes me want to snuggle my cat

anywho

so then would be mulitple the 4 into the (1/2)r^2 sin(theta) equation?

[devout valley]

If you can’t tell, I’m missing some of my troublemakers

[supple plover]

OMG how many kitties do you have?

[devout valley]

And exactly that and if it were a general n instead?

[supple plover]

we would times the n into the equation?

[devout valley]

Well, I don’t actually “have” any cats, but there are a total of 5 that pretty much love to follow and chase me any time I step outside my house

[supple plover]

AWWWWWWWWW

[devout valley]

They’re not even mine but they follow me like they think they are, they even like to sit on my lap too

If I sit down outside then there’s one who’ll just jump up and demand to be petted

[supple plover]

would the equation be n * (1/2) r^2 sin theta?

[supple plover]

AWWW

i wish we had cats roaming around like that

There's too many wild animals around, for there to be kitties..

[devout valley]

Awwww

[devout valley]

(And yep exactly that )

[supple plover]

and then id replace the n with whatever value it is

?

[devout valley]

You would, but we want it in general for our answer, it’s like our recipe

[supple plover]

i see

[devout valley]

This is the answer we wanna give

[supple plover]

ohhh so that equation is the answer

[devout valley]

Yep it is!

[supple plover]

picasso

[devout valley]

Oooh that’s a cool emote I didn’t think I had hehe

[supple plover]

was that also a way to find the answer to number 6?

[supple plover]

its super cute

[devout valley]

Well, not really, but 6 is easy peasy

[supple plover]

okok well we aint there yet, so we can put a pin in it

[devout valley]

We are there we’ve done the area question already haven’t we?

[supple plover]

the question for number 5?

oh we we did

[devout valley]

Yep, now, time for us to get cooking

[supple plover]

🫡

we wouldnt just change n into theta, right? its not that easy is it

[devout valley]

Not that easy, no

[supple plover]

dang..

[devout valley]

Anyways, if you excuse my poor drawing

[supple plover]

no its great 10/10

would buy

[devout valley]

That’s what we have when n is 8

[supple plover]

mhmhmh

[devout valley]

We have n (= 8) copies of those triangles, so how many copies of theta do we have?

[supple plover]

8?

[devout valley]

Yep, that’s it

[devout valley]

So, with my Excellent™️ drawing skills, we see we have 8theta when we go around the center of the circle, right?

[supple plover]

mhmmh i see i see

[devout valley]

And so, what must that 8theta work out as?

(I think at this point, I should be doing the guiding questions )

[supple plover]

LOL

does radians have something to do with it?

[devout valley]

Well, to an extent, we’ll come back to radians in a moment-

[supple plover]

oh okok

[devout valley]

Do you know what the angle at the center of a circle must work out as? Either in degrees or radians, whichever unit is fine for now

[supple plover]

for some reason im wanting to say pi, but i dont believe thats right

180 degrees is the line through the circle

which is also pi

WAIT I CAN COOK

nvm you lost me, i cant find anything about it in my unit circle notes..

i dont think its one of the things my prof went over

[devout valley]

It’s one of those things that you can’t really hint at without giving away the answer

[supple plover]

is it 0

[devout valley]

It is something I know you know, but I don’t know how to ask in a way

[devout valley]

Not quite the answer I’m looking for

If I spin around once, I’ve made a [what] degree turn?

[supple plover]

360

[devout valley]

That’s it

[supple plover]

oh wait, DUH OMG

[devout valley]

It really is one of those things that’s so easy it’s hard, isn’t it

[supple plover]

yea fr

[devout valley]

The 8theta works out as 360 degrees, or 2pi radians

So, as we’re gonna be using radians, as they’re better we have 8theta = 2pi, right?

[supple plover]

mhmhm, quick question how do we know 8 theta is 2pi and not like 4pi

[devout valley]

8theta is the “spinning around once”

[devout valley]

If you look at the picture, you see that they all add up to make the “centre” angle of the circle, which of course is 2pi (because you only go around once)

[supple plover]

mhm i see, so if it was 4theta it would still = 2pi because they all add up to make the "centre" angle of the circle?

[devout valley]

Yep and same for whatever value of n you have

[supple plover]

YAY okay i understand

[devout valley]

Which of course is the general case, n * theta is 2pi

[supple plover]

mhmhm

okay we cooking!

[supple plover]

now we back to this

8theta =2pi

wait is that just the answer for number 6

[devout valley]

Waaaiiittt

[supple plover]

oh wait no

SORRY

[devout valley]

Let’s go here

[supple plover]

okkk

[devout valley]

Don’t be sorry

Remember that we want to express n in terms of theta, that was our aim

[supple plover]

mhmhm

[devout valley]

So if n * theta = 2pi, then what’s n?

[supple plover]

8?

[devout valley]

Not quite what I want

[supple plover]

oh

n is the triangles?

[devout valley]

I’m kinda asking you to rearrange this to make n the subject

[supple plover]

like n = blah blah blah instead?

then would we divide?... so its n= 2pi/theta

yippiii

[devout valley]

Exactly that

Told you we’d get these done

[supple plover]

youre awesome

[devout valley]

You’re more awesome

Now, next question, it’s not too bad either

[devout valley]

If this is the area that we have, but also…

[devout valley]

n is this, can we rewrite the area without an n showing up?

[supple plover]

so then its just r and theta?

[devout valley]

We only want to see r’s and thetas, and no n’s

[supple plover]

mmhmhm so we gotta make a replacement somewhere

im assuming

WAIT

could we just replace the n with the equation?

YIPPII

[devout valley]

Hehe, told you these aren’t too bad

[supple plover]

yup after figuring out all the others it pieces together

[devout valley]

You have to admit that I’m better at the guiding questions then they are, right

[supple plover]

honestly, yes

so much better

[devout valley]

Awwwww

[supple plover]

sadly i have to go back and individually answer the questions as its part of the grade, but thanks to your help itll be a breeze

[devout valley]

Awwww, that’s fair enough I do hope they’ll be fine though, and that you’ll see how to do them, after all, we’d have walked through everything, and we’re so close to done!

[supple plover]

YUP!

luckily thanks to this being on text i can go back too if im stuck on something

OKAY question 8 cooking time

[devout valley]

Yep you can always refer back to what we’ve done, you can search your messages and find what we did

[devout valley]

Yep yep

For that, you notice that the more triangles we start to get, the closer the polygon starts to look like the circle, right?

[supple plover]

yea i see that

[devout valley]

So, the idea is that we wanna take more and more of those sides, making n larger, which then-

Time to make you answer their guided question as n gets larger, what happens to the angle of each of those triangles, the thetas?

[supple plover]

it gets bigger?

turns into infinity?

it continously increases

[devout valley]

The angles don’t get larger, no

[supple plover]

oh

[devout valley]

Think about here, when n = 8

And compare it to, for example, if n = 3 or something

[supple plover]

mhmm

[devout valley]

Or alternatively, remember you have this here!

For both of them, they’re basically telling you something-

[devout valley]

For this one, we can also have theta in terms of n, for which, if n gets larger, what happens to theta?

[supple plover]

whatever value it is, its equal to 360 (2pi)?

[devout valley]

Thats the total of all the angles taken together

[supple plover]

oh i see

[devout valley]

If you had more triangles, you certainly can’t have the angle theta getting bigger, else that would mean you end up getting larger than 2pi, which we don’t want!

[supple plover]

yea yea

[devout valley]

This is also my Perfect™️ representation of when n = 4

[supple plover]

haha 10/10

[devout valley]

Hopefully you notice how the “theta” for the n=4 is bigger than the corresponding one for n = 8, right?

[supple plover]

yea i can visually see it

[devout valley]

Cool, so, finish the sentence with what you think: “as n gets larger, then theta gets [?]”

[supple plover]

idk why im wanting to say smaller, because the triangles are smaller for n = 8

[devout valley]

Why do you not know you’re right

[supple plover]

OH YAYYYYY

[devout valley]

The thetas do get smaller, have some more faith in yourself

[supple plover]

yes teach 🙏

[devout valley]

Another way to see it too is that theta = 2pi/n, and as n goes to infinity, you basically have a fraction with a larger and larger denominator, making the fraction smaller and smaller, decreasing to zero

If you’re happy with that as well?

[supple plover]

yea that also makes sense, but not as much as the picture

so then with this information we write into an expression?

[devout valley]

Pictures are worth a thousand words of course

[supple plover]

very true thats why me likey

[devout valley]

Well, more that you say something like “as n -> infinity, then theta -> 0”

[supple plover]

OHH

tyty